We use the A and X-cluster structures on the Grassmannian to exhibit a polytopal manifestation of mirror symmetry for Grassmannians. From a given cluster seed we obtain both an X-cluster chart and an A-cluster chart for the Grassmannian. We use the X-cluster chart and a naturally defined valuation to construct a Newton-Okounkov body, defined as the convex hull of points. Meanwhile we use the A-cluster to express the superpotential as a Laurent polynomial, and by tropicalizing this expression, we obtain a polytope, defined by inequalities. We prove that the Newton-Okounkov body and the superpotential polytope coincide. In the case that our A-cluster consists of Plucker coordinates, we also give a formula for each lattice point of these polytopes in terms of Young diagrams; using a result of Fulton-Woodward, this formula has an interpretation in terms of quantum cohomology. This is joint work with Konstanze Rietsch.

The tree amplituhedron A(n, k, m) is a geometric object generalizing the positive Grassmannian, which was introduced by Arkani-Hamed andTrnka in 2013 in order to give a geometric basis for the computationof scattering amplitudes in N = 4 supersymmetric Yang-Mills theory. Iwill give an elementary introduction to the amplituhedron, and then describe what it looks like in various special cases. For example, one can use the theory of sign variation and matroids to show that the amplituhedron A(n, k, 1) can be identified with the complex of bounded faces of a cyclic hyperplane arrangement (and hence is homeomorphic to a closed ball). I will also present some conjectures relating the amplituhedron A(n, k, m) to combinatorial objects such as non-intersecting lattice paths and plane partitions.

This is joint work with Steven Karp, and part of it is additionally joint work with Yan Zhang

In joint work with Konstanze Rietsch, we use the cluster structure on the Grassmannian and the combinatorics of plabic graphs to exhibit a new aspect of mirror symmetry for Grassmannians in terms of polytopes. From a given plabic graph G we have two coordinate systems: we have a positive chart for our A-model Grassmannian, and we have a cluster chart for our B-model (Landau-Ginzburg model) Grassmannian. On the A-model side, we use the positive chart to associate a corresponding Newton-Okounkov (A-model) polytope. On the B-model side, we use the cluster chart to express the superpotential as a Laurent polynomial, and by tropicalizing this expression, we obtain a B-model polytope. Our main result is that these two polytopes coincide